Can One Hear the Shape of a Drum?

Kac, Mark

doi:10.1080/00029890.1966.11970915

Cited by 1,051 publications

(852 citation statements)

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“…The result stems from Dikii [4,5], Gelfand [13], and, more directly, from Buslaev and Faddeev [2]; see also Gardner et al [12]. The derivation stems from Kac [25]; see also Kac and van Moerbeke [26]. It employs a more detailed expansion embodied in Besides,…”

Section: Hamiltoniansmentioning

confidence: 99%

“…The simple eigenfunctions are 2 n + 1 in number, so they comprise the whole null space of K. Now for any eigenvalue 2 with eigenfunction f 24 KS=~ ,ci.~_ l X x , qof the function m will now be explained in a series of simple propositions.Proposition 1. m(x, 2) = (2-/t~) ...(2-#o), the auxiliary spectrum i~~ <... < It ~ being computed for the original potential q translated by the amount 0 <-x < 1 so that each #o (i= 1,..., n) is regarded as a function of O< x < 1 25. …”

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confidence: 99%

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The spectrum of Hill's equation

1975

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Let q be an infinitely differentiable function of period t. Then the spectrum of Hill's operator Q = -d2/dx 2 +q(x) in the class of functions of period 2 is a discrete series -.:~.~ < 20 < 21 <)o~ < 23 < 24 <... < '~2i-1 ~ )~2 iT ~'v. Let the number of simple eigenvalues be 2n+ 1 < oc. Borg El] proved that n=0 if and only if q is constant. Hochstadt [21] proved that n= 1 if and only if q = c + 2p with a constant c and a Weierstrassian elliptic function p. Lax [29] notes that n=m if ~ q=4k2K2m(m+l)snZ(2Kx, k). The present paper studies the case n< oo, continuing investigations of Borg Eli, Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardner et al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax[28-30] in various directions. The content may be summed up in the statement that q is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperetliptic irrationality f(~) = IS-(~-,~o) (:~-;,1)...(,~-,~2,).The case n = oo requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [34], Dubrovin and Novikov [6] and A. R. Its and V.B. Matveev [22].

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Section: Hamiltoniansmentioning

confidence: 99%

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confidence: 99%

The spectrum of Hill's equation

1975

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“…In this specific sense, the spectrum σ M encodes information about the characteristic scales of the manifold, while the set of eigenstates B M identifies all the possible diffusion modes, and forms a basis for the algebra of functions on the manifold. Similar considerations can be applied to the problem of wave propagation on the manifold, where the heat equation is replaced by the wave equation; this is the reason behind the famous idea of "hearing the shape of a drum" [30]. The definition of the Laplace-Beltrami operator can be extended easily to more general algebras, like the graded algebra of differential forms or the algebra of functions on a graph [31,32], the latter being of particular importance in our discussion, since, as discussed below, it allows us to implement straightforwardly the spectral analysis on CDT spatial slices, by means of their associated dual graphs.…”

Section: The Laplace-beltrami Operatormentioning

confidence: 99%

Spectrum of the Laplace-Beltrami operator and the phase structure of causal dynamical triangulations

Clemente

D’Elia

2018

Phys. Rev. D

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We propose a new method to characterize the different phases observed in the nonperturbative numerical approach to quantum gravity known as causal dynamical triangulations. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.

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“…Then the functional E(0) := 1 IL (0') 2 (s) ds + P ILIs sin(0s) -0(^)) dt ds (10) 2 2 attains its minimum in F.…”

Section: Structure Of Critical Pointsmentioning

confidence: 99%

“…The minimizer of this problem describes the equilibrium states of a lipid membrane (see, e.g., Arreaga et al [4], and Satake and Honda [13] for its dynamical aspects). In addition, the minimizer presents an example of domains which are determined by the eigenvalues of the Laplace operator, a positive result for Mark Kac's problem [10]; see [16]. Lately, Matsumoto, Murai and Yotsutani [11] studied the structures of critical points of this problem and announced that the critical point of mode n is unique.…”

Section: Introductionmentioning

confidence: 99%

Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application

Watanabe

Izumi

2008

Japan J. Indust. Appl. Math.

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In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic cn( • ) function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.

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Can One Hear the Shape of a Drum?

Cited by 1,051 publications

References 1 publication

The spectrum of Hill's equation

The spectrum of Hill's equation

Spectrum of the Laplace-Beltrami operator and the phase structure of causal dynamical triangulations

Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application

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